This lab will have four separate parts involved: a pre-lab section, day 1, day 2, and a calculation section. Each part will have questions to answer, some writing involved, and some number crunching. You will be responsible for all parts individually, and be required to turn them in together and completed
with a cover sheet. Your grade for the lab will depend on how much you get done, as well as the neatness of the write-up.
Each person in each group must have their own write up.

For the final write-up, you must have all four parts together and in order. It must also be typed with any diagrams not created electronically drawn on the write up page neatly.
If graphs or diagrams are to be included they must also be neat, with labels for
important parts or measurements included. The write-up is designed to be as intensive as a research paper, and will take time to complete. This is why each part is worked on separately, with time to complete the whole write-up at the end. If you do not keep up with each part, it will be very difficult to complete the entire project.

Part One: The Pre-Lab Assessment

Syracuse was a nice Greek town on the Aegean Sea, and Archimedes loved living there. As a matter of fact, this mathematician / physicist loved it so much that he invented machines to defend and improve it. One such machine was the forerunner of what we now know as the catapult. The ‘onager’ (Greek for ‘kicking mule’) was designed to hurl an object (large stones) at the oncoming Roman ships that were attacking the city-state. It employed what is called a torsion spring to store enough energy to fling the stone. It was by no means the first projectile launcher, but it was probably the largest ever built up to that time (about 250 BC).

Archimedes was a very interesting man. He invented many different machines and came up with some important scientific discoveries. Find one of each of these (a machine, and a discovery not mentioned above) that he invented / discovered, and explain both as if to a 12 year-old child.

Explain how you think a catapult works. It is a machine, converting energy from one form to another and performing a task, but how would you explain its workings to someone that can not see or touch it (as if over the phone)?

Why can we classify the marshmallows that we are going to launch as projectiles? What is a projectile, and what is an example of something that is not a projectile (but moves through the air)?

Describe the flight of a marshmallow as it is launched from the catapult, flies through the air, and hits the ground. What would change in its flight, and what would not change? What do we know about its motion that we can calculate?

In order to find out anything about your catapult or how it works, you must take some data. What are three quantities you can measure (collect as data) that will help you to calculate things about your
catapult and how it launches marshmallows? If you get stuck, try looking at some of the equations we use to see what variables might possibly be measurable quantities.

Catapult Lab - Day 1

Each group is to take their catapult, a few marshmallows, and whatever measuring devices they require and meet at a table. This part is mostly about information and getting used to the catapult. Try not to change anything about your catapult, as we want to keep the data as consistent as possible.

Examine the catapult that your group will be using. Take measurements of the important parts (heights, lengths, etc.) and make a diagram of your catapult. A side view with the arm released makes the best schematic diagram. Be sure to remember the number of your catapult in order to maintain consistent data throughout the whole experiment. Also, do not alter the catapult in any way for the same reason.

Try launching a marshmallow from the catapult. Describe what happens to the catapult as you fire it. How might you ‘fix’ the catapult so that more accurate measurements (like launching position) are possible? Describe how the marshmallow is launched (does it leave the catapult before the arm stops, does it roll as it is launched, is it launched at the same firing angle the bucket is pointing). How will any of these be factors in how far or how fast the marshmallow is launched?

One thing you might notice is that the marshmallow leaves the catapult at a certain height above the ground, and lands on ground level. This difference in heights is important to note, as it makes the trajectory of the marshmallow somewhat lopsided. In a simple graph of a marshmallow’s trajectory, draw what I mean. How might you go about fixing this problem; that the marshmallow lands at the same height it was launched from?

Now that the launch height and landing height are the same, we have to consider how we might find the marshmallow’s maximum height (the top of its trajectory). On the same trajectory drawings as in question 3, label the point of maximum height. When the trajectory is fixed so it is not lopsided, where would the maximum height occur in its horizontal motion? Brainstorm any ideas for locating the top of the marshmallow’s path, and being able to measure (not calculate) the maximum height from the launching level.

Since we know about the independence of horizontal and vertical motion, we should be able to calculate the horizontal and vertical velocities separately. This process requires knowing which formulas are for horizontal (non-accelerated) motion and which are for vertical (accelerated) motion. Make a separate list of the equations we know and have used for vertical motion, and one for horizontal motion. Remember that we are assuming only gravity accelerates the marshmallow, and there is not enough air resistance to make any difference.

Now that we have some equations to use, we need some numbers to use them with. Make a table for collecting data in, and try five different shots. We are trying to calculate the horizontal (non-accelerated) velocity of the marshmallow from the time of launch. Be sure to use the method you came up with for equalizing the launch height and landing height.

Calculate the horizontal velocity of the marshmallow for each trial, then find the average. How close is your average to each trial (are there any shots that are drastically different than the average)? What might have caused your trials to differ (sources of error)? If you were to launch the marshmallow with the same horizontal velocity off the edge of a cliff, and it stayed in the air for 5 seconds before hitting the ground, how far away from the cliff (horizontally) would it land on the ground?

Catapult Lab - Day 2

Each group is to take their catapult, a few marshmallows, a stopwatch, meter sticks, and whatever else they need to complete this part of the lab. This part is mostly about data gathering with the catapult. Complete all the trials in each section before moving on to the next part of the experiment. Try not to change anything about your catapult, the marshmallows or how you launch them throughout the lab, as we want to keep the data as consistent as possible.

When writing up this section, you must include a detailed description of how your group tackled each step (including a description of each experiment), answer any questions posed in the lab, and include a separate data table for each part of the experiment. A diagram of each set up is also very helpful.

Now that you know a) how to use the catapult, b) how to have the marshmallow land at the same height as it was launched from, and c) how to picture the marshmallow’s trajectory, it is time to find out more about the catapult using experimental data. First, find a place to shoot from so that you have plenty of room for the marshmallow to fly and land (and not hit anyone or
anything in the process). The hallway is fine, so long as you remember to clean up after yourselves. Once you have reached your spot, set up your catapult so that it is ready to fire. [For each section today, you must keep the catapult in the same spot for every launch.]

Take a few shots in order to get a feel for where the marshmallow will land on a regular basis. Use small pieces of tape to mark the spots where the marshmallow lands for ten shots. (Try to be as accurate as possible when marking the landing spot, and use the smallest pieces of tape that you can.) Once you have all ten shots marked, calculate the distance from the launching point (the catapult) to the center of the tape markings (or nearest to the center as you can get). Now, measure the direct distance between the two most separated pieces of tape. Divide the distance between the two most separated tape pieces by the distance from the catapult to the center of the tape markings. This ratio is called the ‘spread’ of the shots. What does the size of the spread tell you? Would a "better" catapult have a higher or lower value for its spread, and why? Show reasoning with a diagram and an equation.

Remove all the small pieces of tape from the landing site except for the one nearest the ‘average’ landing spot. Be sure to note this average range of your catapult. Now we will try to measure the flight time of the marshmallow. For this part, you will fire 20 trial shots and measure the time of each shot’s flight. Find an average for the set of data, and you have the average flight time of the marshmallow. While watching the marshmallow in mid-air, do you see its trajectory as being affected by air resistance in a noticeable way? How would you expect the trajectory to look if it were? Why are we able to ignore air resistance in our classroom calculations?

For this section, you must try to measure the maximum height achieved by the marshmallow in its trajectory. There are several ways you can do this, but you must decide on the best way among your group and have at least 10 trial shots. You have already brainstormed an idea for this in Day 1, so that might be a good method to try. Be sure to write down what your method was for finding this maximum height, and whether or not you think it was very accurate. If we knew a way to calculate (by way of a formula) the maximum height of the marshmallow using only the flight time, would calculating it be easier and/or more accurate than your method of
measuring the height? Why, or why not?

Catapult Lab - Calculations

Now that all the serious measurements and diagrams have been made, we can now go about analyzing what we found. Prior to performing any real analysis or calculations, you should first try to organize all your information into neat tables and clearly labeled diagrams. It also would help to have attempted an algebraic (no numbers) solution for each of these problems. For each situation that warrants one, you should either have a table of data, a labeled diagram showing what you did, or both.

We will be discussing some of the more complicated calculations in class, so be sure to have read all the parts in order to ask questions about them. You will be responsible for your own calculations, but you can get ideas and corrections from class or your group members. All steps (math work) must be shown towards any solution, and any explanation needed must also be given.

Go around to at least three (3) other groups and find out the spread calculated for their catapults (be sure to note their catapult number). Rank the catapults and tell how yours compares with the other groups. The spread allows us to compare what quality about all the catapults?

We used a meter stick to measure the horizontal distance that the marshmallow traveled, and this method seemed to be rather accurate. When we tried to directly measure the height of the marshmallow’s path, we needed to be more creative, and the measurements were probably not as accurate. We should be able to calculate the maximum height knowing only the total flight time. Calculate this height, and compare it to the measuring-only technique. Which is more accurate, and why?

We already calculated the constant horizontal velocity. How might we go about calculating the initial vertical velocity? We know that the vertical velocity at the top of the trajectory is zero, so we can think of the last half of the flight (from the top of its path to the landing) as being free fall as far as the vertical velocity component is concerned. All we need is half the total flight time, and one equation for vertical (instantaneous) velocity. Calculate the final vertical velocity, and then explain why we can assume that it is equal in size to the initial.

Speed is relative. Would you classify your catapult as having a fast horizontal velocity? How would you say the horizontal velocity compares to the initial vertical velocity of the marshmallow? Draw a vector diagram showing the initial horizontal and vertical components adding together (at a right angle to each other) to form the resultant (or overall) initial velocity.

Draw a trajectory for the marshmallow and include the two initial velocity components (with values) at three points (beginning, top, and landing). Make sure that the horizontal velocity does not change, and the beginning vertical velocity component is equal to (with different direction) the final vertical velocity.

Use your geometrical measurements of the catapult itself to calculate the initial firing angle that the marshmallow leaves the launcher at. This does involve some
trigonometry, and a side-view diagram of the catapult will help enormously.

We can also calculate the firing angle by using our velocity vector diagram. How do your two answers for the firing angle compare? Which is more accurate, and why?

Choose either of the two angles as the "correct" firing angle (as a result of your arguments in question 7), and calculate your percent error for the other
calculated angle. What errors would cause this kind of difference? Perhaps your answers from Day 1, number 2 could shed any light on this.
Overall, what methods could you employ to prevent these, and other errors in
the future?